Related papers: Spin Calogero models and dynamical r-matrices
We demonstrate that in a certain gauge the elliptic Ruijsenaars--Schneider models admit Lax representation governed by the same dynamical $r$--matrix as their non--relativistic counterparts (Calogero--Moser models). This phenomenon was…
Let R be a ring. A construction method for flexible quadratic algebras with scalar involution over R is presented which unifies various classical constructions in the literature, in particular those to construct composition algebras.
We describe classical top-like integrable systems arising from the quantum exchange relations and corresponding Sklyanin algebras. The Lax operator is expressed in terms of the quantum non-dynamical $R$-matrix even at the classical level,…
A constructive procedure is proposed for formulation of linear differential equations invariant under global symmetry transformations forming a semi-simple Lie algebra f. Under certain conditions f-invariant systems of differential…
Spin generalization of the relativistic Calogero-Sutherland model is constructed by using the affine Hecke algebra and shown to possess the quantum affine symmetry $\uqglt$. The spin-less model is exactly diagonalized by means of the…
We establish a correspondence between rational solutions to the matrix KP hierarchy and the spin generalization of the Calogero-Moser system on the level of hierarchies. Namely, it is shown that the rational solutions to the matrix KP…
It is shown that a class of dynamical systems (encompassing the one recently considered by F. Calogero [J. Math. Phys. 37 (1996) 1735]) is both quasi-bi-Hamiltonian and bi-Hamiltonian. The first formulation entails the separability of these…
Textbook treatments of classical mechanics typically assume that the Lagrangian is nonsingular. That is, the matrix of second derivatives of the Lagrangian with respect to the velocities is invertible. This assumption insures that (i)…
The one-dimensional Hubbard model is an exceptional integrable spin chain which is apparently based on a deformation of the Yangian for the superalgebra gl(2|2). Here we investigate the quantum-deformation of the Hubbard model in the…
Differential-difference integrable exponential type systems are studied corresponding to the Cartan matrices of semi-simple or affine Lie algebras. For the systems corresponding to the algebras $A_2$, $B_2$, $C_2$, $G_2$ the complete sets…
We discuss the Poisson structures, Lax matrices, $r$-matrices, bi-hamiltonian structures, the variables of separation and other attributes of the modern theory of dynamical systems in application to the integrable Euler top and to the…
Integrable discretizations are introduced for the rational and hyperbolic spin Ruijsenaars--Schneider models. These discrete dynamical systems are demonstrated to belong to the same integrable hierarchies as their continuous--time…
Quantum Calogero-Moser spin system is a superintegable system with the spectrum of commuting Hamiltonians that can be described entirely in terms of representation theory of corresponding simple Lie group. In this paper the underlying Lie…
A novel algebra underlying integrable systems is shown to generate and unify a large class of quantum integrable models with given $R$-matrix, through reductions of an ancestor Lax operator and its different realizations. Along with known…
The classical $R$-matrix structure for the $n$-particle Calogero-Moser models with (type IV) elliptic potentials is investigated. We show there is no momentum independent $R$-matrix (without spectral parameter) when $n\ge4$. The assumption…
A discrete theory for implicit nonholonomic Lagrangian systems undergoing elastic collisions is developed. It is based on the discrete Lagrange-d'Alembert-Pontryagin variational principle and the dynamical equations thus obtained are the…
We develop algebraic tools for statistical inference from samples of rotation matrices. This rests on the theory of D-modules in algebraic analysis. Noncommutative Gr\"obner bases are used to design numerical algorithms for maximum…
This paper is intended to serve as a review of a series of papers with Nikita Nekrasov, where we achieved several important results concerning the relation between the moduli space of instantons and classical integrable systems. We derive…
The theory of representations of quivers and of their preprojective algebras are reviewed. In particular, moduli spaces of representations of these algebras, quiver varieties and reflection functor are described. The proof that the…
We obtain inductive and enumerative formulas for the multiplicities of the weights of the spin module for the Clifford algebra of a Levi subalgebra in a complex semisimple Lie algebra. Our formulas involve only matrices and tableaux, and…